Our main theorem is the ZH-calculus is complete with respect to its standard interpretation as matrices. That is, if two ZH-diagrams describe the same matrices, then they are provably equal using the rules of the ZH-calculus. Perhaps one of the most appealing features of the calculus is the simplicity of this completeness proof. The core of the proof (section ) fits on 4 pages, where only especially intricate lemmas—which appear in Appendix —were done within the proof assistant Quantomatic [1]}. This is due to two main factors. The first is the extensive use of !-box notation [2]}, which gives an elegant way to reason about diagrams which have arbitrarily-large fan-out-type behaviour. The second is a unique normal form for the ZH-calculus, which expresses any matrix as a Schur product – i.e. entrywise product – of elementary matrices with the property that all but one entry of each matrix is 1.
This multiplicative construction contrasts with the additive construction of the normal form in the ZW-calculus [3]}, which arises as a sum of elementary matrices with the property that all but one entry of each matrix is 0.
For example the normal form of the diagram corresponding to the matrix \(\left({\begin{matrix}a&b\\c&d\end{matrix}}\right)\)  is effectively constructed as follows, where the left-hand side represents the approach in the ZH-calculus with \(*\)  denoting entrywise multiplication, and the right-hand side represents the approach in the ZW-calculus:
\(\begin{pmatrix}a&1\\1&1\end{pmatrix} * \begin{pmatrix}1&b\\1&1\end{pmatrix} * \begin{pmatrix}1&1\\c&1\end{pmatrix} * \begin{pmatrix}1&1\\1&d\end{pmatrix}= \begin{pmatrix}a&b\\c&d\end{pmatrix}= \begin{pmatrix}a&0\\0&0\end{pmatrix} + \begin{pmatrix}0&b\\0&0\end{pmatrix} + \begin{pmatrix}0&0\\c&0\end{pmatrix} + \begin{pmatrix}0&0\\0&d\end{pmatrix}.\) 
